APPENDIX. ON LOGARITHMS. 1. In the solution of trigonometrical problems, the common rules of arithmetic are adequate to almost every case. But the arithmetic operations would be often so laborious, as to be only just within the limits of human industry. In the infancy of mathematics, this difficulty would not have been so severely felt; but, when scientific men began to turn their attention to subjects of natural philosophy, they found their progress impeded by the tedious calculations which were absolutely necessary to be performed. Under this embarrassment, they naturally turned their thoughts to the invention of some means of evading the difficulty. After several essays, John Napier, a Scotch Baron, hit upon the method of Logarithms. His invention received a most important improvement from Henry Briggs, a Fellow of St. John's College, Cambridge, and Professor of Geometry in the University of Oxford. To explain the nature of these Logarithms, and how to calculate them, is the object of this treatise. The introductions to the tables will afford the student rules for using them and examples, of their utility. 2. DEF. If u = ax, x is called the logarithm of u to the base a, and is thus denoted, loga u.* 3. By assuming different values of the base, there will be as many different systems of logarithms. No systems, however, are used, except the Napierien and Briggean. The base of the former is and is always called e; that of the latter is 10. The Briggean, from their nature, are often called decimal; and tabular, because they are used in arithmetic operations, and therefore inserted in the tables. They are of such frequent use, that they are denoted simply by log, instead of log10 4. PROP. In every system the logarithm of the base is unity. 5. PROP. .. a = a .. loga = 1. In every system the logarithm of unity is zero. 6. COR. The logarithm is positive, or negative, according as the number is greater or less than unity. *Mathematicians are indebted for this notation to Mr. Jarrett, of Catherine Hall, Cambridge. 7. PROP. In every system the logarithm of zero is an infinitely great quantity, negative, or positive, according as the base is greater or less than unity. Loga(U1. Uş. Uz... Un) = loga u1 + loga u2+ loga uz+...+loga un' 2° .. loga (U1. U2.U.....un) = loga u1+loga u2+loga uz+..+loga un• и 2 (log+log +log uloga u). 11. PROB. To explain the particular advantages of the decimal logarithms. In this system, the logarithm of 10 is unity. Hence, the logarithms of numbers between 10 and 0 are fractions less than unity. These are placed in the tables. And, since log 10m u= log 10m + log u = m + log u, the logarithms of all other numbers may be found by the addition or subtraction of the quantity m, which is called the characteristic. For example, the tables give This would not be the case if any other number were the base, for then the logarithm of 10m would not be m. The finding other logarithms of numbers would thus become a laborious operation, unless the tables were increased in size, by inserting the separate logarithms. And for the inverse operation of finding the number from the logarithm this increase would evidently become absolutely necessary. It is, therefore, chiefly for the latter reason that Briggs' alteration of the base from e to 10 is so great an improvement. This will obtain whatever be the value of y. therefore, by division, Let y = 0, 13. COR. 1. In the Napierien logarithms, the base, which is always called e, is assumed of such a value, that (e−1)— (e− 1 )2 + † (e− 1)3 — ... .. log, u(u-1) — (u — 1)2 + † (μ − 1 ) 3 — ... SS |